Radioactivity: (i) half-life of an isotope; m t 1/2 = ln

The idea

Half-life t₁/₂ is the average time taken for half the nuclei in a sample to decay, or for the activity to halve. It is a constant for a given isotope. The decay constant λ (sometimes written m) is related to half-life by λ t₁/₂ = ln 2, where ln 2 ≈ 0.693. This equation allows conversion between λ and t₁/₂. For example, if t₁/₂ = 5.0 years, then λ = ln 2 / 5.0 ≈ 0.139 year⁻¹. Half-life can range from fractions of a second to billions of years. In calculations, ensure time units are consistent. The equation is used in problems involving activity, number of nuclei and decay constant.

What earns the marks

• Half-life t₁/₂ is the time for half the undecayed nuclei in a sample to decay, or for activity to halve.
• The decay constant λ (or m) is related to half-life by λ t₁/₂ = ln 2.
• Ln 2 is the natural logarithm of 2, approximately 0.693.
• Using the equation to calculate λ from t₁/₂ or t₁/₂ from λ, with correct units.
• Half-life is constant for a given isotope and independent of sample size or external conditions.

Common mistakes and how to put them right

• Using the wrong time unit.

  Put it right: Convert all times to the same unit before calculating.

• Confusing half-life with mean lifetime.

  Put it right: Use λ t₁/₂ = ln 2 for half-life.

• Forgetting that half-life is the time for half the nuclei to decay, not for all to decay.

  Put it right: After each half-life, the remaining fraction halves.

• Misreading the equation as λ t₁/₂ = ln (without 2) or omitting ln 2.

  Put it right: The correct equation is λ t₁/₂ = ln 2.

Exam tips

• In multiple-choice questions, check that the equation is rearranged correctly before substituting numbers.
• Keep ln 2 to at least three significant figures (0.693) for accurate calculations.
• If a question gives activity or count rate, remember that half-life is the time for these to halve as well.